Randomization time

Figure 19.8.5 displays the graph of tliis cdf. Anodier example of a cdf of a continuous random time variable is shown in Figure 19.8.6. A cdf of a continuous variable (a normal distribudon - to be reviewed in the next chapter) is provided in Figure 19.8.7.

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Finally, if a Markov process is in some state 5 at some (reference) time t0, and if 6 is the random time required for the process to leave state s to go to some other state, then the probability that 6 is larger than some arbitrary time t is given by... 

Light is emitted from the bulk material at random times and in all directions, such that the photons emitted are out of phase with each other in both time and space. Light produced by spontaneous emission is therefore called incoherent light. 

Sometimes random sampling is difficult to execute, as when a stream is being monitored with a time-activated automatic remote sample collection device. Under such conditions a random start or other superimposed random time element may be substituted. The efficiency of systematic sampling improves as the population becomes better understood. Both theoretical and experimental studies of this point have been made (14). 

Experimental data shows a strong variation of the effective temperature with the waiting time by several orders of magnitude. The voltage signal is also intermittent with strong voltage spikes at random times. The distribution of the... 

In chromatography the quantitative or qualitative information has to be extracted from the peak-shaped signal, generally superimposed on a background contaminated with noi%. Many, mostly semi-empirical, methods have been developed for relevant information extraction and for reduction of the influence of noise. Both for this purpose and for a quantification of the random error it is necessary to characterize the noise, applying theory, random time functions and stochastic processes. Four main types of statistical functions are used to describe the tosic properties of random data ... 

Exercise. A cathode emits electrons at independent random times. Derive for the spectral density of the current fluctuations... 

It is a random walk over the integers n = 0,1,2,... with steps to the right alone, but at random times. The relation to chapter II becomes more clear by the following alternative definition. Every random set of events can be treated in terms of a stochastic process Y by defining Y(t) to be the number of events between some initial time t = 0 and t. Each sample function consists of unit steps and takes only integral values n = 0,1, 2,... (fig. 5). In general this Y is not Markovian, but if the events are independent (in the sense of II.2) there is a probability q(t) dt for a step to occur between t and t + dt, regardless of what happened before. If, moreover, q does not depend on time, Y is a Poisson process. 

---